Problem: Takumi plants a tree in his backyard and studies how the number of branches grows over time. He predicts that the relationship between $N$, the number of branches on the tree, and $t$, the elapsed time, in years, since the tree was planted can be modeled by the following equation. $N=5 \cdot 10^{0.3t}$ According to Takumi's model, in how many years will the tree have $100$ branches? Give an exact answer expressed as a base- $10$ logarithm. years
Thinking about the problem We want to know how many years, $t$, it will take for the number of branches on the tree, $N$, to reach $100$. So we need to find the value of $t$ for which $N=100$. Substituting $100$ in for $N$ in the model gives us the following equation. $100=5 \cdot 10^{0.3t}$ Solving the equation We can solve the equation as shown below. $\begin{aligned}5\cdot 10^{0.3t}&=100\\\\ 10^{0.3t}&=20\\\\ 0.3t&=\log\left(20\right)\\\\ t&=\dfrac{{\,\log\left(20\right)}}{0.3}\\\\ \end{aligned}$ It will take $\dfrac{{\,\log\left(20\right)}}{0.3}$ years for the tree to have $100$ branches. The expression above represents an exact solution to the problem. We can use a calculator to approximate the value of the expression, but this will be a rounded inexact answer. The answer The answer is $\dfrac{{\,\log\left(20\right)}}{0.3}$ years.